>If he claims that the system he invents is a ring, then he is not telling the truth

He (Martinez) didn't, so what is it? Keep addition as is, but be willing to change the usual rules of multiplication in order to impose the new sign rule: (-) x (-) = (-).

By the way, Cardano's investigations were labeled "deviant" math by some, and at least one author accused him of possessing a "deviant psychology", so invitations to these sorts of investigations must come with appropriate warning labels. The motivation for Cardano seems to have been trying to invent an algebra that admits negative numbers and roots without leading to imaginary numbers.